ODE
\[ b x \left (a^2-x^2\right ) y'(x)^2+\left (a^2-x^2\right ) y'(x)^3-b x-y'(x)=0 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Dependent variable missing, Solve for \(y'\)
Mathematica ✓
cpu = 0.0314726 (sec), leaf count = 64
\[\left \{\left \{y(x)\to c_1-\frac {b x^2}{2}\right \},\left \{y(x)\to \tan ^{-1}\left (\frac {x}{\sqrt {a^2-x^2}}\right )+c_1\right \},\left \{y(x)\to c_1-\tan ^{-1}\left (\frac {x}{\sqrt {a^2-x^2}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.032 (sec), leaf count = 52
\[ \left \{ y \left ( x \right ) =-{\frac {b{x}^{2}}{2}}+{\it \_C1},y \left ( x \right ) =-\arctan \left ( {x{\frac {1}{\sqrt {{a}^{2}-{x}^{2}}}}} \right ) +{\it \_C1},y \left ( x \right ) =\arctan \left ( {x{\frac {1}{\sqrt {{a}^{2}-{x}^{2}}}}} \right ) +{\it \_C1} \right \} \] Mathematica raw input
DSolve[-(b*x) - y'[x] + b*x*(a^2 - x^2)*y'[x]^2 + (a^2 - x^2)*y'[x]^3 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -(b*x^2)/2 + C[1]}, {y[x] -> ArcTan[x/Sqrt[a^2 - x^2]] + C[1]}, {y[x] -> -ArcTan[x/Sqrt[a^2 - x^2]] + C[1]}}
Maple raw input
dsolve((a^2-x^2)*diff(y(x),x)^3+b*x*(a^2-x^2)*diff(y(x),x)^2-diff(y(x),x)-b*x = 0, y(x),'implicit')
Maple raw output
y(x) = -1/2*b*x^2+_C1, y(x) = arctan(x/(a^2-x^2)^(1/2))+_C1, y(x) = -arctan(x/(a^2-x^2)^(1/2))+_C1